
where \(\omega \)  is the angular frequency, \(k\)  the wavenumber, the \(c_n\)  are some order one coefficients (whose precise value is not important here) and we have ignored suppressed \(H\) -dependent corrections, since \(H^2/\omega ^2 \ll 1\)  for the frequencies relevant here. Here we explicitly write \(c_{\rm GW}\)  as a function of \(k\)  and hence denote the asymptotic value of \(c_{\rm GW}\)  for low energies/frequencies, i.e. effectively its cosmological value derived in (REF ), by \(c_{\rm GW}(0)\)  – see [1]} for details on the derivation of the full dispersion relation for this example.
Crucially, (REF ) asymptotes to the cosmological \(c_{\rm GW}= c_{\rm GW}(0)\)  for small \(k\)  (low frequencies) and asymptotes to luminal \(c_{\rm GW}= c\)  for large \(k\)  (high frequencies). Here it is important to emphasise the point made by [1]} that asymptoting to \(c_{\rm GW}= c\)  at high energies, as is entailed by (REF ), is a generic consequence as long as Lorentz invariance is restored at scales \(\gtrsim M\) . So this feature should certainly not be seen as an accident of the example interactions shown here.
